184_notes:examples:week2_electric_potential_positive

Example: Electric Potential from a Positively Charged Balloon

Suppose we have a positively charged balloon with total charge $Q=5.0\cdot 10^{-9} \text{ C}$ . What is the electric potential (also called voltage) at a point $P$ , which is a distance $R=50 \text{ cm}$ from the center of the balloon?

Facts

The balloon has total charge $Q=5.0\cdot 10^{-9} \text{ C}$ .
The point $P$ is a distance $R=50 \text{ cm}$ away from the center of the balloon.
The electric potential due to a point charge can be written as $V = \frac{1}{4\pi\epsilon_0}\frac{q}{r},$ where $q$ represents the charge and $r$ is the distance.

Representations

Assumption

We assume $P$ lies outside of the balloon. We make this assumption because it was not specified, but this seems to make more sense than $P$ being inside the balloon. This also helps us draw the representation below, which can be used to bolster our approximation later on of the balloon as a point charge.

Charged Balloon, and Point P

Goal

Find the electric potential at $P$ .

Solution

Approximation

We approximate the balloon as a point charge. We do this because we have the tools to find the electric potential from a point charge. This seems like a reasonable approximation because the balloon is not too spread out, and we are interested in points outside the balloon. so the balloon might “look” like a point charge from the perspective of an observation location that is little far away.

Assumption

The electric potential infinitely far away from the balloon is $0 \text{ V}$ . Read here for why this is important.

The electric potential at $P$ is given by

$\begin{align*} V &= \frac{1}{4\pi\epsilon_0}\frac{q}{r} \\ &= \frac{1}{4\pi\cdot 8.85\cdot 10^{-12} \frac{\text{C}}{\text{Vm}}}\frac{5\cdot 10^{-9} \text{ C}}{0.5 \text{ m}} \\ &= 90 \text{ V} \end{align*}$